In this paper we prove a compactness property for sequences of discrete functions defined on Poisson point clouds. We show (Theorem REF ) that a sequence with uniformly bounded \(p\) -Dirichlet non-local energy admits a converging subsequence, in a suitable topology (see Definition REF ). Non local energies on discrete systems have been intensively treated, under the variational point of view, in the last two decades, as a way of modelling several phenomena: from fracture mechanics to image denoising and crystallization. Typical discrete variational energies have their natural definition on regular or stochastic lattices (see for instance [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]} and references therein). In some situations we need to account in the model for a randomness component. This leads towards a natural extension of classical models on lattices to a more general point cloud framework. Among other interesting features, variational energies on point clouds have been proposed also as a tool to implement supervised or semi-supervised learning algorithms, clustering problem, data denoising and image denoising (as a partial list of literature on the topic we refer the reader to [14]}, [15]}, [16]}, [17]}, [18]} and references therein).
